# Proofs for the derivatives of eˣ and ln(x) – Advanced differentiation

In this article, we are going to cover the proofs of the derivative of the functions **ln(x)** and **e ^{x}**. Before proceeding there are two things that we need to revise:

### The first** principle of derivative**

Finding the derivative of a function by computing this limit is known as differentiation from first principles. Derivative by the first principle refers to using algebra to find a general expression for the slope of a curve. It is also known as the **delta method**. The derivative is a measure of the instantaneous rate of change, which is equal to

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**e in terms of limit**

The number e, known as** Euler’s number**, is a mathematical constant approximately equal to **2.71828**. The discovery of the constant itself is credited to Jacob Bernoulli in 1683 who attempted to find the value of the following expression (which is equal to e).

**Proof for the Derivative of e**^{x}

^{x}

### Example 1: Find the derivative of ?

**Solution:**

By the chain rule,

### Example 2: Find the derivative of ?

**Solution:**

Use here the quotient rule:

**Proof for the Derivative of ln(x)**

### Example 1: Find the derivative of 3ln(x)?

**Solution:**

3ln(x)’ = 3(1/x) = 3/x

### Example 2: Find the derivative of ln(x)/5?

**Solution:**

(ln(x)/5)’ = 1/5(ln(x))′ = (1/5) (1/x) = 1/5x